Laws of radiation of heat Heated Bodies Radiate
We shall now turn to another puzzle confronting physicists at the turn of the century -- just how heated bodies radiate. There was a general understanding of the mechanism involved -- heat was known to cause the molecules and atoms of a solid to vibrate, and it was known that the molecules and atoms were themselves complicated patterns of electrical charges. From the experiments of Hertz and others Maxwell's predictions that oscillating charges emitted electromagnetic radiation had been confirmed, at least for simple antennas. It was known from Maxwell's equations that this radiation traveled at the speed of light and from this it was realized that light itself, and the closely related infrared heat radiation, were actually electromagnetic waves. The picture, then, was that when a body was heated, the consequent vibrations on a molecular and atomic scale included some oscillating charges. If one assumed that Maxwell's theory of electromagnetic radiation, which worked well in the macroscopic world, was also valid at the microscopic scale (tenths of nanometers), then these oscillating charges would radiate, presumably giving off heat and light.
What is meant by the phrase "black body" radiation? The point is that radiation from heated bodies depends to some extent on the body heated. Let us back up momentarily and consider how differently different materials absorb radiation. Some, like glass, seem to absorb light hardly at all-the light goes through. For a shiny metallic surface, the light isn't absorbed either, it gets reflected. For a black material like soot, light and heat are almost completely absorbed, and the material gets warm. How can we understand these different behaviors in terms of light as an electromagnetic wave interacting with charges in the material, causing them to oscillate and absorb energy from the radiation? In the case of glass, evidently this doesn't happen, at least not much. A full understanding of how this works needs quantum mechanics, but the general idea is as follows. There are charges-electrons-in glass that are able to oscillate in response to an applied external oscillating electric field, but these charges are tightly bound to atoms, and only oscillate at certain frequencies. It happens that for ordinary glass none of these frequencies correspond to those of visible light, so there is no resonance with a light wave, and hence little energy absorbed. Glass is opaque at some frequencies outside the visible range (in general, both in the infrared and the ultraviolet). These are the frequencies at which the electrical charge distribution in the atoms or bonds can naturally oscillate.
A piece of metal has electrons free to move through the entire solid. This is why metals can conduct electricity. It is also why they are shiny. These unattached electrons oscillate together with large amplitude in response to the electrical field of an incoming light wave. They themselves then radiate electromagnetically, just like a current in an antenna. This radiation from the oscillating electrons is the reflected light. In this situation, little of the incoming radiant energy is absorbed, it is just reradiated, that is, reflected.
Soot, like a metal, will conduct an electric current, although not nearly so well. There are unattached electrons, which can move through the whole solid, but they keep bumping into things-they have a short mean free path. When they bump, they cause vibration, like a pinball machine, so they give up energy into heat. Although the electrons in soot have a short mean free path compared to that in a good metal, they move very freely compared with electrons in atoms, so they can accelerate and pick up energy from the electric field in the light wave. They are therefore effective intermediaries in transferring energy from the light wave into heat.
Absorption and Emission
Heated bodies radiate by processes just like the absorption described above operating in reverse. Thus, for soot heat causes the lattice to vibrate more vigorously, giving energy to the electrons (imagine them as balls in a pinball machine with strongly vibrating barriers, etc.) and since the electrons are charged they radiate away excess kinetic energy. On the other hand, the electrons in a metal have very long mean free paths, the lattice vibrations affect them much less, so they are less effective in radiating away heat. It is evident from considerations like this that good absorbers of radiation are also good emitters.
At sufficiently high temperatures, all bodies become good radiators. Items heated until they glow in a fire look much more similar than they do at room temperature. For a metal, this can be understood in terms of a shortening of the mean free path by the stronger vibrations of the lattice interfering with the electron's passage.
Observing the Black Body Spectrum
Any body at any temperature above absolute zero will radiate to some extent, but the intensity and frequency distribution of the radiation depends on the detailed structure of the body. To make any progress in understanding radiation, we must specify the details of the body radiating. The simplest possible model is to consider a body which is a perfect absorber, and therefore the best possible emitter. For obvious reasons, this is called a "black body".
A good model absorber is a small hole in the side of a box. Radiation impinging on the hole from outside enters the box and is absorbed as it is scattered around inside, so only a tiny fraction is re-emitted.
We shall, therefore, take a cubical box (an oven, in other words) with a small hole in the side as our model to analyze emission of radiation. Fortunately, this is a system which can also be investigated experimentally fairly easily. A beam of radiation from the hole in the oven is passed through a diffraction grating and projected on to a screen, where it is separated out by wavelength (or equivalently frequency). A detector is moved up and down along the screen to find how much radiant energy is being emitted in each frequency range, say the energy between frequency f and f + df is RT(f)df.
We can then plot RT(f) against f for a given temperature T. It is found experimentally that for small f, RT(f) is proportional to f2, a parabolic shape, but for increasing f it falls below the parabola, peaks at fmax, then falls quite rapidly to zero as f increases beyond fmax.
Suppose we now double the absolute temperature - how does that affect RT(f)?
For those low frequencies where RT(f) was parabolic, doubling the temperature doubles the intensity of the radiation. However, it is found that at 2T the curve follows the (doubled) parabolic path much further - in fact, twice as far: fmax(2T) = 2fmax(T).
The curve R2T(f), then, reaches eight times the height of RT(f). It also spreads over twice the lateral extent, so the area under the curve, corresponding to the total energy radiated, increases sixteenfold on doubling the temperature. This is Stefan's Law of Radiation: the power P radiated from one square meter of black surface at temperature T:
Note that the graph above plots the energy intensity in the oven. This energy is being pumped out of the opening at speed c. When allowance is made for the fact that an opening of unit area is effectively less than unit area for radiation coming in at an angle, and half the energy is in waves moving in directions away from the hole, the actual rate of emission of energy through a hole of unit area is equal to (1/4)c times the energy intensity inside. The blue curve in the graph above is for 10,000K. The energy intensity plotted on the y-axis is in joules/m3 per unit frequency range, and it is evident on glancing at the graph that integrating to give the total intensity one will find something around 8 joules/m3. To find the heat energy radiated per second per square meter of surface at 10,000K, we multiply this energy density by c/4, which gives around 600 Megawatts per square meter.
The upward shift in fmax with T is something everyone is familiar with - when an iron is heated in a fire, the first visible radiation (at around 900K) is deep red - the lowest frequency visible light. Further increase in T causes the color to change to orange then yellow, and white at a very high temperature, signifying that all the visible frequencies are being emitted roughly equally.
The change in fmax is linear in the absolute temperature;
This is called Wein's Displacement Law.
For example, the sun's surface temperature is 5700K, and at that temperature much of the energy is radiated as visible light. This is no accident - evolution has adapted us to see most efficiently in the light most readily available.
This shift in the frequency at which radiant power is a maximum is very important for harnessing solar energy, such as in a greenhouse. We need glass which will allow the solar radiation in, but not let the heat radiation out. This is feasible because the two radiations are in very different frequency ranges - 5700K and, say, 300K - and there are materials transparent to light but opaque to infrared radiation. This is only possible because fmax varies with temperature.
Understanding the Black Body Spectrum: the Analogy with a Gas
Experimental investigations of the radiation intensity as a function of temperature and frequency, RT(f) above, give well-defined reproducible results, not very sensitive to the material of the oven, etc. For a small hole, the radiation emitted will be a representative sample of the radiation bouncing around inside the box.
In fact, the general shape of the function RT(f) was very reminiscent of the energy distribution in a gas in thermal equilibrium, as analyzed by Maxwell and Boltzmann. In contrast to the molecules in a gas, the different wavelengths of radiation do not collide with each other in the middle of the oven. Nevertheless, energy can shift from one mode to the other through the intermediary of the oscillators in the wall - which is to say, just the atoms and molecules the wall is made of. An electromagnetic wave in the oven can set a charge oscillating in the wall, energy can be transferred to other charges in the wall, which will in general oscillate and radiate at different frequencies. Thus the different frequencies of radiation inside the oven will come to thermal equilibrium.
Wilhelm Wien was born on January 13, 1864 at Fischhausen, in East Prussia. He was the son of the landowner Carl Wien, and seemed destined for the life of a gentleman farmer, but an economic crisis and his own secret sense of vocation led him to University studies. When in 1866 his parents moved to Drachstein, in the Rastenburg district of East Prussia, Wien went to school in 1879 first at Rastenburg and later, from 1880 till 1882, at the City School at Heidelberg. After leaving school he went, in 1882, to the University of Göttingen to study mathematics and the natural sciences and in the same year also to the University of Berlin. From 1883 until 1885 he worked in the laboratory of Hermann von Helmholtz and in 1886 he took his doctorate with a thesis on his experiments on the diffraction of light on sections of metals and on the influence of materials on the colour of refracted light. His studies were then interrupted by the illness of his father and, until 1890, he helped in the management of his father's land. He was, however, able to spend, during this period, one semester with Helmholtz and in 1887 he did experiments on the permeability of metals to light and heat rays. When his father's land was sold he returned to the laboratory of Helmholtz, who had been moved to, and had become President of, the Physikalisch-Technische Reichsanstalt, established for the study of industrial problems. Here he remained until 1896 when he was appointed Professor of Physics at Aix-la-Chapelle in succession to Philipp Lenard. In 1899, he was appointed Professor of Physics at the University of Giessen. In 1900 he became Professor of the same subject at Würzburg, in succession to W.C. Röntgen, and in this year he published his Lehrbuch der Hydrodynamik (Textbook of hydrodynamics).